Emergent fractal phase in energy stratified random models

Abstract

We study the effects of partial correlations in kinetic hopping terms of long-range disordered random matrix models on their localization properties. We consider a set of models interpolating between fully-localized Richardson’s model and the celebrated Rosenzweig-Porter model (with implemented translation-invariant symmetry). In order to do this, we propose the energy-stratified spectral structure of the hopping term allowing one to decrease the range of correlations gradually. We show both analytically and numerically that any deviation from the completely correlated case leads to the emergent non-ergodic delocalization in the system unlike the predictions of localization of cooperative shielding. In order to describe the models with correlated kinetic terms, we develop the generalization of the Dyson Brownian motion and cavity approaches basing on stochastic matrix process with independent rank-one matrix increments and examine its applicability to the above set of models.

Circular Rosenzweig-Porter random matrix ensemble

“…It would be interesting to see if similar generalization could be constructed for the circular RP ensemble, next to other Floquet models with multifractal eigenstates [47][48][49][50]. This might for example be achievable by considering stochastic processes with correlated increments, which has been initialized for the RP ensemble recently [67]. Second, the circular RP ensemble could potentially be of value in studies on random quantum circuits [68] as a non-maximally random building block, analog to e.g.…”

Non-ergodic delocalized phase with Poisson level statistics

“…The above exponential (58) and the power-law (61) decay result in the piecewise linear f (α), Eq. (34). The exponential decay with the prefactor 1/ξ corresponds to the plateau at r ξ, where |ψ…”

“…However going to the nonergodic (especially MBL) phase, we face an immediate problem as the overwhelming majority of the random-matrix ensembles shows multifractal properties in the Hilbert space only at the very point of the Anderson transition [26]. There are only few ensembles like the Rosenzweig-Porter model [27] or some related ones [28][29][30][31][32][33][34][35][36] which provide an entire phase of non-ergodic delocalized eigenstates. However, all such models show the standard Wigner-Dyson level repulsion in the whole non-ergodic phase, like in Gaussian random matrix ensembles 2 .…”

“…This localization, symmetric with respect to a = d, survives also under the positional disorder[50] and finite anisotropy[51], however it is fragile to the reduction of correlations[29,34] and mutual disorientation of different dipoles[44,45] …”

“…This method has recently been generalized in[36] in order to describe non-ergodic delocalized states as well. Alternative methods have been considered in[34].…”